Memory in Classical and Quantum systems

In a classical system the trajectroy of a particle under an invere square law force is typically the result of solving the equation m r''(t) =const/r^2. By integrating this eqaution we can see that the r'(t) at any instant depends on all the history of that system till this instant. Can we consider this a memory effect? Then what is the difference between it and the evolution equation governing the a two level system interacting with an (infinte) bath like the field modes of vacuum? In studying this phenomenon in Weisskopf Wigner theory we had to solve a similar equation for Ca(t), the upper state probability amplitude, and neglect the effect of the history of the system on Ca'(t). (See for example eq. 14.102 in "Laser Physics" by scully et. al.) Even then we can say that the equation Ca'(t)=Gamma/2 Ca(t) depends on the history simply because we have to include initial conditions.So in general when can we say that a system is Markovian (doesn't have a memory ) and when are we able to neglect this memory ?